Edexcel GCE Core Mathematics C1 June may 2011 exam answers review

Find the value of

1. 25^½(1)
2. 25^-3/2(2)

Given that y = 2x⁵ + 7 + 1/x³, x≠0 find, in their simplest form,

1. dy/dx(3)
2. \(\int\)y dx4)

The points P and Q have coordinates (–1, 6) and (9, 0) respectively.

The line l is perpendicular to PQ and passes through the mid-point of PQ.

ind an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

(5)

Solve the simultaneous equations

(7)

A sequence a₁, a₂ , a₃ ,… is defined by

where k is a positive integer.

1. Write down an expression for 2 a in terms of k.(1)
2. Show that a₃ = 25k +18.(2)
3. • Find [IMAGE] in terms of k, in its simplest form.
   • Show that [IMAGE] is divisible by 6.

   (4)

Given that 6x + 3x^5/2/√x can be written in the form 6x^p + 3x^q,

1. write down the value of p and the value of q.(2)

Given that dy/dx = 6x + 3x^5/2/√x, and that y = 90 when x = 4,

2. find y in terms of x, simplifying the coefficient of each term. (5)

where k is a real constant.

1. Find the discriminant of f (x) in terms of k.(2)
2. Show that the discriminant of f (x) can be expressed in the form (k + a)² + b, where
   a and b are integers to be found.(2)
3. Show that, for all values of k, the equation f (x) = 0 has real roots.(2)

Figure 1 shows a sketch of the curve C with equation y = f (x).

The curve C passes through the origin and through (6, 0).

The curve C has a minimum at the point (3, –1).

On separate diagrams, sketch the curve with equation

1. y = f (2x),(3)
2. y = −f (x),(3)
3. y = f (x + p), where p is a constant and 0<p<3.(4)

On each diagram show the coordinates of any points where the curve intersects the x-axis and of any minimum or maximum points.

1. Calculate the sum of all the even numbers from 2 to 100 inclusive,
   2 + 4 + 6 + …… + 100

   (3)

2. In the arithmetic series
   k + 2k + 3k + …… + 100

   k is a positive integer and k is a factor of 100.

   1. Find, in terms of k, an expression for the number of terms in this series.
   2. Show that the sum of this series is
      50 + 5000/k

      (4)

   3. Find, in terms of k, the 50th term of the arithmetic sequence

      (2k + 1), (4k + 4), (6k + 7), …… ,

      giving your answer in its simplest form.

      (2)

The curve C has equation

1. Sketch C, showing the coordinates of the points at which C meets the axes.(4)
2. Show that dy/dx = 3x² + 14x + 15(3)

The point A, with x-coordinate -5, lies on C.

3. Find the equation of the tangent to C at A, giving your answer in the form y = mx + c, where m and c are constants. (4)

Another point B also lies on C. The tangents to C at A and B are parallel.

4. Find the x-coordinate of B.(3)